Full Salpeter Equation with Confining Interactions: Analytic Stability Proof

TL;DR

This paper provides an analytic proof of the stability of the Salpeter equation with confining interactions, addressing known instabilities and clarifying the spectral properties of bound states in quantum field theory.

Contribution

It offers the first rigorous stability proof for the Salpeter equation with confining kernels, clarifying spectral properties and resolving issues related to Klein's paradox.

Findings

Eigenvalues are real and discrete for stable bound states.

Spectrum includes real eigenvalues and pairs of opposite sign, plus imaginary points.

Stability is proven for time-component vector harmonic-oscillator kernels.

Abstract

The most popular 3-dimensional reduction of the Bethe-Salpeter formalism for the description of bound states within quantum field theory is the Salpeter equation, found as the instantaneous limit of the Bethe-Salpeter framework if allowing, in addition, for free propagation of the bound-state constituents. Unfortunately, depending on the Lorentz nature of the Bethe-Salpeter kernel, supposedly stable results of Salpeter's equation with confining interactions exhibit (un-)expected instabilities, probably related to Klein's paradox. Clearly, bound states may be regarded as stable if, for appropriate interactions, their mass eigenvalues belong to a real and discrete (part of the) spectrum that is bounded from below. Some general features of the eigenvalue spectra of any Salpeter equation are well-established: all bound-state masses squared are real and, for a large class of sufficiently reasonable Bethe-Salpeter kernels - which includes all interactions of phenomenological relevance for QCD -, the spectrum of mass eigenvalues consists, in the complex bound-state mass plane, at most of real pairs of opposite sign and imaginary points. For time-component vector harmonic-oscillator kernels, a straightforward stability proof is given.